The completeness in Theorem REF  is relative to systems of
size \(s\)  which is a consequence of the fact that we synthesize first-order
recurrence systems. That is, there exists a system of recurrence equations of order \({}>1\)  and
size \(s\)  with an algebraic relation \(p\in \mathbb {K}[x_1,\dots ,x_s]\) , but there exists no
first-order system of size \(s\)  where \(p\)  is an algebraic relation. Therefore, in
order to be able to synthesize a system satisfying \(p\)  as an algebraic relation, we have to look for recurrence systems of size \(>s\) . Example REF  next
discusses an instance of such a higher-order recurrence system.
In order to gain completeness in Theorem REF 
independent of the size \(s\)  of the recurrence system, we would need to be able to derive an upper bound on \(s\) . Obtaining such a bound from \(p\)  seems to be highly
non-trivial with connections to the inverse problem of difference Galois theory [1]}, and is subject of future work.
